tppreqb - Metamath Proof Explorer

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Theorem tppreqb 4035

Description: An unordered triple is an unordered pair if and only if one of its elements is a proper class or is identical with one of the another elements. (Contributed by Alexander van der Vekens, 15-Jan-2018.)

**Assertion**
Ref	Expression
tppreqb	$/\$ $/\$

**Proof of Theorem tppreqb**
Step	Hyp	Ref	Expression
1		3ianor 982	. . . 4 $/\$ $/\$ $\/$ $\/$
2		df-3or 966	. . . 4 $\/$ $\/$ $\/$ $\/$
3	1, 2	bitri 249	. . 3 $/\$ $/\$ $\/$ $\/$
4		orass 524	. . . . 5 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
5		ianor 488	. . . . . . . 8 $/\$ $\/$
6		tpprceq3 4034	. . . . . . . 8 $/\$
7	5, 6	sylbir 213	. . . . . . 7 $\/$
8		tpcoma 3992	. . . . . . 7
9		prcom 3974	. . . . . . 7
10	7, 8, 9	3eqtr3g 2498	. . . . . 6 $\/$
11		orcom 387	. . . . . . . 8 $\/$ $\/$
12		ianor 488	. . . . . . . 8 $/\$ $\/$
13	11, 12	bitr4i 252	. . . . . . 7 $\/$ $/\$
14		tpprceq3 4034	. . . . . . 7 $/\$
15	13, 14	sylbi 195	. . . . . 6 $\/$
16	10, 15	jaoi 379	. . . . 5 $\/$ $\/$ $\/$
17	4, 16	sylbi 195	. . . 4 $\/$ $\/$ $\/$
18	17	orcs 394	. . 3 $\/$ $\/$
19	3, 18	sylbi 195	. 2 $/\$ $/\$
20		df-tp 3903	. . . 4
21	20	eqeq1i 2450	. . 3
22		ssequn2 3550	. . . 4
23		snssg 4028	. . . . . . 7
24		elpri 3918	. . . . . . . 8 $\/$
25		nne 2626	. . . . . . . . . 10
26		3mix2 1158	. . . . . . . . . 10 $\/$ $\/$
27	25, 26	sylbir 213	. . . . . . . . 9 $\/$ $\/$
28		nne 2626	. . . . . . . . . 10
29		3mix3 1159	. . . . . . . . . 10 $\/$ $\/$
30	28, 29	sylbir 213	. . . . . . . . 9 $\/$ $\/$
31	27, 30	jaoi 379	. . . . . . . 8 $\/$ $\/$ $\/$
32	24, 31	syl 16	. . . . . . 7 $\/$ $\/$
33	23, 32	syl6bir 229	. . . . . 6 $\/$ $\/$
34		3mix1 1157	. . . . . . 7 $\/$ $\/$
35	34	a1d 25	. . . . . 6 $\/$ $\/$
36	33, 35	pm2.61i 164	. . . . 5 $\/$ $\/$
37	36, 1	sylibr 212	. . . 4 $/\$ $/\$
38	22, 37	sylbir 213	. . 3 $/\$ $/\$
39	21, 38	sylbi 195	. 2 $/\$ $/\$
40	19, 39	impbii 188	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 184 $\/$ wo 368 $/\$ wa 369 $\/$ w3o 964 $/\$ w3a 965

wceq 1369

wcel 1756

wne 2620

cvv 2993

cun 3347

wss 3349

csn 3898

cpr 3900

ctp 3902

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1591 ax-4 1602 ax-5 1670 ax-6 1708 ax-7 1728 ax-10 1775 ax-11 1780 ax-12 1792 ax-13 1943 ax-ext 2423

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1372 df-ex 1587 df-nf 1590 df-sb 1701 df-clab 2430 df-cleq 2436 df-clel 2439 df-nfc 2577 df-ne 2622 df-v 2995 df-dif 3352 df-un 3354 df-in 3356 df-ss 3363 df-nul 3659 df-sn 3899 df-pr 3901 df-tp 3903

This theorem is referenced by: (None)

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